On $\sigma$-local formations of finite groups with bounded $\mathfrak{H}_\sigma$-defect

I. N. Safonova ; V. V. Skrundz

Problemy fiziki, matematiki i tehniki, no. 1 (2025), pp. 87-101

Résumé

Let $\mathfrak{F}$ and $\mathfrak{H}$ be some $\sigma$-local formations of finite groups. Then $\mathfrak{F}/_\sigma\ \mathfrak{H}\cap\mathfrak{F}$ denote the lattice of all $\sigma$-local formations $\mathfrak{X}$ such that $\mathfrak{H}\cap\mathfrak{F}\subseteq\mathfrak{X}\subseteq\mathfrak{F}$. The length of the lattice $\mathfrak{F}/_\sigma\ \mathfrak{H}\cap\mathfrak{F}$ is called a $\mathfrak{H}_\sigma$-defect of the $\sigma$-local formation $\mathfrak{F}$. In particular, if $\mathfrak{H}$ is the formation of all identity groups, then the $\mathfrak{H}_\sigma$-defect of a $\sigma$-local formation $\mathfrak{F}$ is called a $l_\sigma$-length of the formation $\mathfrak{F}$. The general properties of $\mathfrak{H}_\sigma$-defect of $\sigma$-local formations are studied, the description of minimal $\sigma$-local non-$\mathfrak{H}$-formations for an arbitrary $\sigma$-nilpotent $\sigma$-local formation $\mathfrak{H}$ is obtained, the description of the lattice structure of $\sigma$-local formations of $\mathfrak{H}_\sigma$-defect $1$ is given. The descriptions of the lattice structure of reducible $\sigma$-local formations of finite $\mathfrak{H}_\sigma$-defect, as well as the lattice structure of reducible $\sigma$-local formations of finite $l_\sigma$-length are obtained.

Détail

Keywords: finite group, $\sigma$-local formation, critical $\sigma$-local formation, $\mathfrak{H}_\sigma$-defect of a $\sigma$-local formation, $l_\sigma$-length of a $\sigma$-local formation.